quantum monte carlo simulations of solid 4whitlock/lsscs06pub.pdf · quantum monte carlo...

Quantum Monte Carlo Simulations of Solid 4He

P.A. Whitlock1 and S.A. Vitiello2

1 Computer and Information Science Department,Brooklyn College, CUNY Brooklyn, NY 11210, USA

[email protected] Instituto de Fısica, Universidade Estadual de Campinas,

13083 Campinas — SP, [email protected]

Abstract. Recent experimental investigations [20] of solid 4He have beeninterpreted as showing possible superfluidity in the solid at low tempera-tures, below 0.2 K. A solid behaving this way, exhibiting both long rangetranslational order and superfluidity, has been called a supersolid phase.The existence of a supersolid phase was proposed many years ago [1], andhas been discussed theoretically. In this paper we review simulations of thesolid state of bulk 4He at or near absolute zero temperature by quantumMonte Carlo techniques. The techniques considered are variational cal-culations at zero temperature which use traditional Bijl-Dingle-Jastrowwavefunctions or more recently, shadow wavefunctions; Green’s functionMonte Carlo calculations at zero temperature; diffusion Monte Carlo, andfinally, the finite temperature path integral Monte Carlo method. A briefintroduction to the technique will be given followed by a discussion of thesimulation results with respect to solid helium.

1 Introduction

After many years of investigation, the properties of the solid phases of bulk, 4He,were felt to be well understood [3]. At temperatures near absolute zero, 4He existsin both the solid and fluid states. The crystalline structure is known to exhibithexagonal closest packing (hcp) rather than the face centered cubic packing ex-pected for a three-dimensional hard sphere system. Even at absolute zero, theatoms exhibit zero point motion around their lattice positions which leads to avery “loose” solid at densities near melting. In the 1960’s, Andreev and Lifshitz[1] proposed that such quantum solids could sustain superfluidity. One indicatorof the onset of superfluidity is the Bose-Einstein condensate, the fraction of theatoms condensed into the zero momentum state. The condensed atoms acquirequantum mechanical coherence over macroscopic length scales. Quantum MonteCarlo simulations observed a Bose-Einstein condensate of several percent in liq-uid 4He systems [34]; but only detected a condensate in a quantum solid whenthe atoms were interacting with a Yukawa potential [6].

Interest has been recently renewed by the torsional oscillator experiments ofKim and Chan [20]. Ultrahigh-purity 4He was confined in a torsion cell andsubjected to pressures between 26 and 66 bars to reach the solid phase. A non-classical rotational inertia fraction that can be associated with superflow was

I. Lirkov, S. Margenov, and J. Wasniewski (Eds.): LSSC 2005, LNCS 3743, pp. 40–52, 2006.c© Springer-Verlag Berlin Heidelberg 2006

Quantum Monte Carlo Simulations of Solid 4He 41

observed at temperatures below 230 milliKelvin. These observations have leadto renewed interest in the measurements of the Bose-Einstein condensate andother measures of superfluidity in simulations of the properties of solid helium.

4He systems may be studied theoretically by solving the appropriateSchrodinger or Bloch equation. At absolute zero, the behavior of an N bodyhelium system is described by the eigenfunction of the Schrodinger equation in3N dimensional space:

[−∇2 + V (R)]Ψ0(R) = E0Ψ0(R) (1)

where R ≡ ri | i = 1, . . . , N and the ri are the positions of the individ-ual atoms. V(R) represents the interaction potential between the atoms in thesystem. The term [−∇2 + V (R)] is the Hamiltonian, H, for the system and iswritten here in dimensionless form. Knowledge of the physical relationships be-tween the atoms can be built into a parametrized mathematical form for a trialwavefunction, ΨT (R), an approximation to Ψ0(R). The variational energy can beminimized with respect to the parameters through the Monte Carlo evaluationof the expectation value of the ground energy, E0. This technique is referredto as variational Monte Carlo (VMC). Approaches where the simulation resultsare subject only to statistical uncertainties are referred to as Quantum MonteCarlo (QMC) methods. In the Green’s function Monte Carlo (GFMC) methodthe integral transform of the Schrodinger equation, (1), is iterated by performinga random walk in the configuration space of the N atoms to yield an asymptot-ically exact solution. Such a solution can also be obtained by sampling a shorttime Green’s function followed by an extrapolation of the results to account forthe time step errors introduced by the approximation. This technique is knownas diffusion Monte Carlo (DMC). Finally, finite temperature systems may bestudied by considering the Bloch Equation:

[−∇2 + V (R) + ∂/∂t]B(R, t) = 0, (2)

where B(R, t) is the many-body density matrix. Path integrals [12, 16] MonteCarlo (PIMC) simulations can be performed and for small enough temperatureintervals, the density matrix can be compared to the ground state eigenfunctionfrom (1).

In the following sections, the results of applying the techniques outlined aboveto the simulation of solid 4He will be described.

2 Variational Monte Carlo Methods

Given a trial wavefunction, ΨT (R), an estimator for the variational energy,

ET =〈ΨT |H |ΨT 〉〈ΨT |ΨT 〉 ≥ E0, (3)

is an upper bound to the true ground state energy E0 of the system. The lowestvariational energy is obtained through a minimization process with respect tothe parameters in ΨT (R). In the coordinate representation (3) becomes:

42 P.A. Whitlock and S.A. Vitiello

ET =∫

dRΨ∗T (R)HΨT (R)

∫dR|ΨT (R)|2 =

∫dRπ(R)EL(R), (4)

where dR ≡ d3r1d3 . . . rN . The last term of the above equation, EL(R), the local

energy, is obtained by multiplying and dividing the numerator of (3) by ΨT (R),

EL(R) =HΨT (R)ΨT (R)

, (5)

and π(R) is a normalized probability distribution function,

π(R) =|ΨT (R)|2

∫dR|ΨT (R)|2 . (6)

2.1 Trial Wavefunctions

As point out by Feenberg [11] a plausible general form for the exact ground-statewavefunction of a system of N interacting bosons is

Ψ(R) =∏

i<j

f2(rij)∏

i<j<k

f3(i, j, k)∏

i<j<k<l

f4(i, j, k, l) · · · (7)

= exp12

⎡

⎣∑

i<j

u(rij) +∑

i<j<k

u3(i, j, k) +∑

i<j<k<l

u4(i, j, k, l)

⎤

⎦ · · · (8)

In the liquid phase, the simplest variational function, the so called Bijl-Dingle-Jastrow or Jastrow trial function, considers only a single term of the aboveexpression: u2(rij). The first computer simulation for a system of helium atomswas performed by McMillan [22] using u2(r) = b/r5 and reasonable results wereobtained. A better approximation to the variational wavefunction which includedthree-body correlations [28], u3(i, j, k), led to an improvement of about 10% inthe estimated values of the energies.

For the solid phase, the usual approach was to write the trial wavefunction as

ψTsol(R) =∏

i<j

f2(rij)Φ(R), (9)

where Φ(R) is a model function, ideally, a permanent of localized single particleorbitals. However, since the effect of the quantum statistics on the energy isminor, Φ(R) is left unsymmetrized. Gaussian orbitals were used in simulationsperformed by Hansen and Levesque [14] with reasonable results. The inclusion oftriplet correlations, f3(i, j, k), in the trial wavefunction lead to an improvementof about 15% in the simulations results [32]. The introduction of higher-order cor-relations in a trial wavefunction has become feasible by introducing the shadowwavefunction [30], discussed more fully below.

The functional form of correlation factors in a trial wavefunction can be fullyoptimized. The idea is to write the correlation factors as a sum of the elementsof a basis set [31]. For the two-body correlation factor,


f(r) =M∑

n=1

cnfn(r), (10)

where the fn are functions of the basis set and the cn are variational parameters.If the basis is well chosen, a small value of M is sufficient to recover all theenergy associated with the correlation under consideration. For (10), a verysuitable basis is the one obtained by solving for the lowest M energy states ofthe Schrodinger-like equation involving a pair of helium atoms,

(

− h

2m∇2 + V (r)

)

fn(r) = λnfn(r). (11)

The boundary conditions are such that at a distance d, chosen as a cutoff oreventually as a variational parameter, the functions fn go smoothly to 1 or toa function that gives the correct long range behavior of the system. One of theadvantages of this method is to automatically obtain an optimal correlation atsmall values of r. Since, the wavefunction is small when r → 0, it is difficult tosample this very important region of configuration space. Thus, the usual MonteCarlo optimization of the trial function, does not perform well at small r.

2.2 Monte Carlo Techniques

The simulation starts by sampling the normalized probability distribution π(R)of (6), i. e., by constructing a sequence of points Ri|i = 1, . . . , M in theconfiguration space. More formally we require [19] that Ri belong to the sequencewith probability given by

PrRi =∫

Ω

dRπ(R) (12)

for any Ω ⊂ Ω0 of the sample space Ω0. The sampling in most cases is performedusing the Metropolis [23] algorithm.

If we consider M independent samples, the variational energy is estimated as

EM =1M

M∑

i=1

EL(Ri). (13)

In the limit of large M we have EM → ET . This energy is obtained without anyuncontrolled approximations or nonconvergence for any form of the wavefunctionand is subject only to statistical uncertainties of O(M−1/2). The statistical erroris easily estimated. Variance reduction techniques, e. g. importance sampling, canbe used to reduce the multiplicative factor that appears in the calculation of theerror. Other properties of the system can also be readily estimated.

2.3 Variational Results on Solid 4He

The earliest variational calculations could not differentiate between a crystalwith fcc packing and one with hcp order. Chester [10] showed that a Jastrow


wavefuntion, ΨT (R) =∏

i<j f(rij), supports a Bose-Einstein condensate. How-ever, this pair product form without the localization provided by φ(r), as in (9),only crystallizes at a very high density [15]. Therefore the use of gaussian one-body orbitals, φ(r), was introduced. This, however, precluded the observation ofa Bose-Einstein condensate.

Recently, Vitiello considered in great detail the question of the ground-statestructure of solid helium using the most recent he-he potential of interaction.Performing careful variational calculations and employing reweighting, he wasable to show that the hcp order is favored in the 4He system [29].

2.4 Shadow Wavefunction Calculations

The construction of trial functions based on the inclusion of auxiliary variables,“shadow particles”, is a very successful approach within the variational methods.These trial functions are particular representations of the Feenberg form [11],where one is able to introduce tractable correlations up to the number of particles

ΨSh(R) =∏

i<j

f(rij)∏

i<j<k

f(3)ijk . . .

∏

i<j...<w

f(N)ij...w. (14)

The variational shadow wavefunction is defined in terms of an integral overauxiliary variables S ≡ si|i = 1, . . . , N in the whole space

ΨSh(R) =∫

Ξ(R, S)dS, (15)

where Ξ is a function that includes a factor of the Jastrow form dependent solelyon the configuration space coordinates R, a Gaussian coupling between the spacevariables ri and the auxiliary variables si, and a term of the Jastrow form thatcorrelates the si among themselves:

Ξ(R, S) = exp

⎛

⎝−∑

i<j

12

(b

rij

)5

−∑

i

C|ri − si|2 −∑

i<j

γV (δsij)

⎞

⎠ . (16)

In this formulation ΨSh(R) depend on the He-He interacting potential and fourvariational parameters: b, C, γ and δ. Since the auxiliary variables, due to thelast term of Eq. (16), are isomorphic to the coordinates of a system of particlesinteracting through V , we call the auxiliary variables shadow particles.

A trial wavefunction that can correlate all the atoms in the system is impor-tant by itself. In addition, there are strong physical motivations to deal withsuch a class of variational wavefunctions: Feynman’s path integrals in imaginarytime and justifications from projection methods.

Shadow wavefunctions have enabled the investigation of disorder phenomenain solid 4He such as vacancies [24] or the interfacial region between a solid anda liquid at coexistence [25] by variational calculations. This is possible becausewith the shadow wavefunction approach both the fluid phase and solid phases canbe described, without the introduction of single particle orbitals. Moreover theBose symmetry is manifestly maintained and so relaxation around non-localizeddefects are allowed.


3 Green’s Function Monte Carlo

If the potential energy in (1) is bounded from below V (R) ≥ −V0, (1) can berewritten as:

[−∇2 + V (R) + V0]Ψ0(R) = (E0 + V0)Ψ0(R). (17)

A Green’s function,

[−∇2 + V (R) + V0]G(R, R0) = δ(R − R0) (18)

can be derived with the same boundary conditions as Ψ0(R) and used to trans-form (17) into an integral equation:

Ψ0(R) = (E0 + V0)∫

G(R, R′)Ψ0(R′)dR′. (19)

Since the ground-state wavefunction and Green’s function for a Bose system arenon-negative; the ground state wavefunction and approximations to it may betreated as probability distribution functions. The Green’s function may also beused as a distribution function for R conditional on the previous position R′.The integral version of the Schrodinger equation, (19), is solved by a Neumanniteration starting with a zeroth order approximation, such as a trial wavefunctionoptimized in a variational calculation. A population of points R′ is sampledfrom ΨT (R′) and a new set of points R is sampled from (Et + V0)G(R, R′)where Et is an approximation to the ground-state energy. As this process isrepeated, at the nth iteration the set of points R′ has been sampled from ψ(n)

and the next generation of points, n+1, is sampled from:

ψ(n+1)(R) = (Et + V0)∫

G(R, R′)ψ(n)(R′)dR′. (20)

Equation (20) defines one step of a random walk whose asymptotic distributionis Ψ0(R). Since the simulation system is composed of N atoms with periodicboundary conditions, the Schrodinger equation has a discrete spectrum and theiterations are guaranteed to converge.

The procedure may be made computationally more efficient and the variancereduced by employing an importance sampling transformation [18]. Let ΨT (R) bea trial wavefunction which may be the same as ψ(0)(R) and Ψ(R) = ΨT (R)Ψ(R),then (19) becomes

Ψ(R) = (E + V0)∫

[ΨT (R)G(R, R′)/ΨT (R)]Ψ(R′)dR′. (21)

The sequence of functions obtained by iteration of the integral equation con-verges to ΨT (R)Ψ0(R) and Et is chosen such that the random walk is stable.

Unfortunately, the Green’s function, (18), is not known analytically owing tothe complexity of the boundary conditions. However, to implement the algorithmrepresented by (20) or (21), it is not necessary to know the full Green’s function;


it is sufficient to develop a method to sample configurations from G(R, R′). TheGreen’s function may be written as,

G(R, R′) =∫ ∞

0G(R, R′, τ)dτ (22)

G(R, R′, τ) is the Green’s function for a Bloch equation, (2),

(H + ∂/∂τ)G(R, R′, τ) = δ(R − R′)δ(τ). (23)

For a given configuration, R0, a finite domain, D(R0), may be chosen such thatthe potential of interaction, V(R), is bounded from above within the domain bythe constant, U(R0). A domain Green’s function may then be defined:

[−∇2 + U(R1) + ∂/∂τ ]GU (R1, R0, τ) = δ(R1 − R0)δ(τ) (24)

subject to the boundary condition that GU (R1, R0, τ) = 0 whenever R1 is onthe boundary or outside of D(R0). Physically, (24) represents a diffusion processof a particle in a domain with a constant absorption rate and a perfectly ab-sorbing boundary. Multiplying (18) by GU (R1, R0, τ) and (24) by G(R, R1, τ),integrating both equations over R1 and subtracting the two resulting equationsyields:

G(R, R0, τ) = GU (R, R0, τ)

+∫

∂D(R0)

(∂GU (R1, R0, τ − τ0)

∂n

)

G(R, R1, τ0)dR1 (25)

+∫

D(R0)(U(R0) − V (R1))GU (R1, R0, τ − τ0)G(R, R1, τ0)dR1

Equation (26) is a linear integral equation for G(R, R0, τ) in terms of GU (R, R0,τ). In the second term of the right hand side of (26), the boundary conditionfor GU (R1, R0, τ) has been used to convert a volume integral into a surfaceintegral over ∂D(R0) and the derivative normal to the domain’s surface, ∂/∂nhas been introduced. The domain, D(R0) may be chosen in any convenient way;in particular, as the Cartesian product of three-dimensional spheres or cubescentered at R0. GU (R, R0, τ) is known analytically and may be interpreted as aconditional probability distribution function. Thus, points R may be sampledby a random walk governed by (26) for any given set R0.

An asymptotically unbiased estimator for the energy is given by

Em =∫

Ψn(R)HΨT (R)dR∫

Ψn(R)ΨT (R)dR. (26)

Except for statistical sampling and convergence errors, (26) is an exact estima-tor for the ground state energy. For other properties of the physical system,F(R), that do not commute with the Hamiltonian, a “mixed” estimator may bedefined as

〈F 〉m =∫

Ψn(R)F (R)ΨT (R)dR∫

Ψn(R)ΨT (R)dR. (27)


If the trial wavefunction ΨT (R) is “close” to the actual ground state wave-function, ΨT (R) = Ψ0(R) + δψ(R), then a linear extrapolation may be used toestimate the exact value to within an order δ2:

〈F 〉x = 2〈F 〉m − 〈F 〉T (28)

where 〈F 〉T is the variational value calculated with ΨT (R). It was shown [34] thatthis extrapolation process gave the same expected value as the random walkbased on the “forward walking” algorithm but with much smaller statisticalerrors. The extrapolated value was also shown to be independent of the trialwavefunction used.

As in variational calculations, the result of the GFMC simulations is a wave-function represented as an ensemble of configurations of atomic positions.Through (27) and (28), the Bose-Einstein condensate fraction can be obtainedfor the helium system. The fraction of particles in the zero-momentum state isgiven by the asymptotic limit of n(r),

n0 = limr→∞n(r). (29)

The one-body density matrix, n(r), is a measure of the change in the wavefunc-tion for given displacement r and is the fourier transform of the momentumdistribution, n(k):

n(r) =∫

eik·rn(k)dk

=⟨

Ψ(r1, r2, · · ·, ri + r, · · ·, rn)Ψ(r1, r2, · · ·, ri, · · ·, rn)

⟩

. (30)

The first calculation of the Bose-Einstein condensate in solid 4He using the GFMCmethod [7] involved the trial wavefunction of Eq. (9) and the Lennard-Jones po-tential of interaction [4]. It was of course recognized that the form of the trialwavefunction that was used as an importance function might bias the results and,not surprisingly, the condensate fraction was less than 1%. Additional calculations[34], showed that the Lennard Jones potential itself was inadequate to describe thehelium system. A further investigation of the Bose-Einstein condensate concludedthat Ceperley, et. al. [6] observed a condensate because they used a translationallyinvariant wavefunction for the importance function. The GFMC simulations wererepeated using a more realistic form for the potential of interaction, the HFDHE2potential [2] and improved wave-functions [17]. However, no condensate fractionwithin statistical error was observed in the solid phase [35, 36].

In a variational calculation using a shadow wavefunction, when the densityof a system of helium atoms reaches the appropriate value, a state with trans-lationally broken symmetry is spontaneously produced. Thus, it was natural tointroduce shadow wavefunctions as importance functions in GFMC calculations.In order to perform these calculations, Whitlock and Vitiello [33] made an ex-tension to the GFMC method such that the shadow degrees of freedom whereupdated using the Metropolis algorithm according to the probability distribu-tion of (16). Despite good results for some of the properties of the system, the


variance of the calculation did not encourage further attempts to compute thecondensate fraction in the solid phase. However the idea of using the shadowwavefunction ideas in a QMC method seems promising.

3.1 Diffusion Monte Carlo

The time dependent Schrodinger equation in imaginary time t → it/h,

∂ψ(R, t)∂t

= −(−∇2 + V (R) − Et)ψ(R, t), (31)

is equivalent to the classical diffusion equation with sources represented by V (R).In (31), the Hamiltonian H has been written as the sum of the kinetic energyT , −∇2, plus the potential energy V (R) displaced by a trial energy Et, whichdoes not change the description of the state of the system.

In a short time approximation, the Green’s function for (31) can be writtento O(t) as,

G(R, R′, δt) ≈ 〈R|e−V (R)δt/2e−Tδte−V (R′)δt/2eEtδt|R′〉. (32)

It is possible to rewrite the above expression as the product of a rate term,

w(R, R′, δt) = exp(−(V (R) + V (R′))δt

2+ Etδt), (33)

times a propagator, identified as the Green’s function for ordinary diffusion,

Gd(R, R′, δt) = 〈R|e−Tδt|R′〉 = (4πδt)−3N/2 exp(

− (R − R′)2

4δt

)

. (34)

In a simulation, for each R′ in a given set of configurations, a new R is easilysampled from Gd and weighted by w(R, R′, δt). By repeating these steps andperforming a suitable extrapolation to t → 0, the results will yield an estimateof the ground-state energy if Et ≈ E0. This is shown by writing the formalsolution of the time dependent Schrodinger equation as

ψ(R, t) =∑

i

ϕi(R)e−i(Ei−Et)t/h, (35)

where the ϕi(R) are an orthogonal basis set.The method presented so far is very inefficient due to the branching process

and because the random walk may explore unimportant regions of the configura-tion space. Here again an importance sampling transformation as in (21) allowsthe simulations to converge faster and more efficiently. If (31) is multiplied by atrial wavefunction ψT , it can be written in the coordinate representation as

∂ψ(R, t)∂t

= −(−∇2 + ∇ · F(R) + F(R) · ∇ − (Et − EL(R)

)ψ(R, t), (36)

where ψ(R, t) = ψT (R)ψ(R, t), EL(R) is the local energy and F(R) = 2∇ lnψT (R). If we compare the above expression with equation (31), it still includes


a branching process, given by V = EL(R) − Et. The diffusion process has asuperimposed drift velocity given by the two last terms of the expression, −∇2+∇ · F(R) + F(R) · ∇.

By taking the short time approximation, as before we can write:

G(R, R′, δt) = W (R, R′, δt)Gd(R, R′, δt), (37)

where

W (R, R′, δt) = exp(

−(EL(R) + EL(R′))δt

2+ Etδt

)

, (38)

and

Gd(R, R′, δt) = (4πδt)−3N/2 exp(

− (R − R′ − δtF(R))2

4δt

)

. (39)

Simulations that include importance sampling converge to ψT ψ. Instead of com-puting V (R), now we calculate EL which approaches a constant as ψT (R) goesto the true eigenfunction of the system. This is important since the simulationsbecome much more stable. Moreover, the drift guides the random walk to theimportant regions of the configuration space.

4 Path Integral Monte Carlo

All static and, in principle, dynamic properties of a many-body quantum system,such as 4He, at thermal equilibrium, may be obtained from the density matrix,(R, R′, β), the solution to the Bloch equation, (2). β represents an inverse tem-perature or “imaginary time”, β = 1/kT . The solution to the Bloch equationcan be written in the coordinate representation as:

(R, R′, β) =< R|e−βH |R′ > (40)

For distinguishable particles, the density matrix is non-negative for all valuesof its arguments and can be interpreted as a probability distribution function.If two density matrices are convoluted together, a density matrix at a lowertemperature results:

< R|e−(β1+β2)H |R′ >=∫

< R|e−β1H |R′′ >< R′′|e−β2H |R′ > dR′′. (41)

The integral over R′′ may be evaluated using a generalization of the Metropo-lis sampling algorithm [26, 8]. Starting at a sufficiently high temperature, thedensity matrix may be accurately written as an expansion in one and two-bodydensity matrices. Then, multiple convolutions can be performed to reduce thetemperature to near absolute zero.

The density matrix for a boson system such as 4He is obtained from thedistinguishable particle density matrix by using the permutation operator toproject out the symmetric component,

< R|e−(βH)|R′ >B=

∑℘ < ℘R|e−(βH)|R′ >

N !(42)

The sum over permutations is performed by a Monte Carlo technique.


To calculate the momentum distribution requires obtaining the off-diagonalparts of the density matrix. In one method, an atom is displaced off the diagonalby a distance r while the other atoms and permutation are held fixed while n(r),(30), is computed. This method is very accurate at small r. In a second method, oneatom is again off the diagonal, but the distance between the two ends of the path forthat atom is allowed to vary. This allows the calculation of n(r) at large r [9].

Ceperley and Bernu [5] have found that the superfluid density observed inPIMC simulations of solid 4He are strongly affected by the size of the systemsimulated. A 48 atom system exhibits a 1.2% superfluid density at 55 bars pres-sure while a 180 atom systems has zero superfluid density. They conclude thatthe phenomenon observed by Kim and Chan can not be explained by vacanciesor interstitials in the equilibrium bulk 4He system.

4.1 Path Integral Ground State Calculations

Ground state expectations values at finite temperatures can be efficiently cal-culated by using a path integral ground state Monte Carlo method [27]. Theintegral equation in imaginary time equivalent to the Schrodinger equation is

ψ(R, t) =∫

G(R, R′, t − t0)ψ(R′, t0)dR′. (43)

In the above equation, G(R, R′, t) is the propagator of (23). As was seen in theprevious section, this propagator is viewed as density matrix operator corre-sponding to an inverse temperature β and simulated by path integrals.

The difference in the present method compared to PIMC is that a truncatedpath on a trial wavefunction is considered instead of periodic boundary condi-tions in imaginary time as the trace of G(R, R′, t) requires. Since the groundstate eigenfunction can be obtained by filtering a suitable trial function ψT

ψ0(R) = limt→∞ψ(R, t) = lim

t→∞

∫G(R, R′, t)ψT (R′)dR′, (44)

the ground sate expectation value of any operator can be written as

〈O〉 =〈ψt|G(t)OG(t′)|ψT 〉〈ψt|G(t)G(t′)ψT 〉 . (45)

If the convolution of the density matrix of (41) is divided into N time steps,β/N = t/N = δt,

G(R, R′, t) =∫

dR1dR2 · · · dRN−1ρ(R, R1, δt)ρ(R1, R2, δt)ρ(RN−1, R′, δt)

(46)and substituted in (45), we obtain

〈O〉 =

∫ ∏Ni=0 dRiO(RM )ψT (R0)

(∏N−1i=0 ρ(Ri, Ri+1, δt)

)ψT (RN )

∫ ∏Ni=0 dRiψT (R0)

(∏N−1i=0 ρ(Ri, Ri+1, δt)

)ψT (RN )

, (47)


where R0 = R, RM = R′ and RM is an internal time slice. For a convergedcalculation, if the operator is placed on the extreme edges of the path one getsa mixed estimator. If RM is in the middle, the exact expectation value of theground state is obtained. The paths are sampled using the Metropolis algorithm.Samples that do not include coordinates of the trial wavefunction are performedas in PIMC.

Galli and Reatto [13] have employed this formalism using the shadow wave-function as ΨT (R) to study confined solid 4He. Their model system contains alarge static spherical object that uses a purely repulsive potential to prevent thehelium atoms from reaching the center of the simulation cell. As a consequenceof the periodic boundary conditions this correspond to a static lattice of hardcore spheres. They observe that the freezing pressure increases to about 38 atm.This behavior is comparable to that found in experiments of 4He confined invycor [21]. In addition they observe that the disorder induced by the mismatchbetween the 4He crystalline structure and the static hard spheres induces delo-calization. This is a necessary condition to have off-diagonal long range order inthe system. Also, the presence of a Bose-Einstein condensate requires delocaliza-tion of the atoms. These results could be relevant in explaining the observationof a supersolid phase for 4He in vycor [21].

However, to date, no quantum Monte Carlo studies have observed delocaliza-tion or Bose-Einstein condensation in the pure bulk solid 4He.

References

1. A.F. Andreev and L.M. Lifshitz: Soviet Phys. JETP 29 (1969) 11072. R.A. Aziz, V.P.S. Nain, J.S. Carley, W.L. Taylor, and G.T. McConville: An accu-

rate intermolecular potential for helium. J. Chem. Phys. 70 (1979) 4330–43423. K.H. Bennemann and J.B. Ketterson, eds.: The Physics of Liquid and Solid Helium.

Wiley, New York (1976)4. J. de Boer and A. Michels: Contribution to the Quantum Mechanical Theory of

the Equation of State and the Law of Corresponding States. Determination of theLaw of Force of Helium. Physica (Utrecht) 5 (1938) 945–957

5. D.M. Ceperley and B. Bernu: Ring Exchanges and the Supersolid Phase of 4He.Phys. Rev. Lett. 93 (2004) 155303-1–4

6. D.M. Ceperley, G.V.Chester, and M.H.Kalos: Monte Carlo study of the ground stateof bosons interacting with Yukawa potentials. Phys. Rev. B 17 (1978) 1070–1081

7. D.M. Ceperley, G.V. Chester, M.H. Kalos, and P.A. Whitlock: Monte Carlo Studiesof Crystalline Helium. Journal de Physique 39 Colloque C6 (1978) 1298–1304

8. D.M. Ceperley and E.L. Pollock: Path-integral computation of the low-temperatureproperties of liquid 4He. Phys. Rev. Lett. 56 (1986) 351–354

9. D.M. Ceperley and E.L. Pollock: The momentum distribution of normal and su-perfluid liquid 4He. Can. J. Phys. 65 (1987) 1416

10. G.V. Chester: Speculations on Bose-Einstein Condensation and Quantum Crystals.Phys. Rev. A 2 (1970) 256–258

11. E.Feenberg: Ground state of an interacting boson system. Ann. Phys. (N.Y.) 84(1974) 128

12. R.P. Feynman: The lambda-Transistion in Liquid Helium. Phys. Rev. 90 (1953)1116–1117


13. D.E. Galli and L.Reatto: The shadow path integral ground state method: study ofconfined solid 4He. J.Low Temp. Phys. 136 (2004) 343–359

14. J.P. Hansen and D. Levesque: Ground state of solid helium–4 and –3. Phys. Rev.165 (1968) 293–299

15. J.P. Hansen and E.L. Pollock: Ground-State Properties of Solid Helium-4 and -3.Phys. Rev. A 5 (1972) 2651–2665

16. M. Kac: Probability and Related Topics in Physical Science. Interscience, NewYork (1959)

17. M.H. Kalos, M.A. Lee, P.A. Whitlock, and G.V. Chester: Modern potentials andthe properties of condensed 4He. Phys. Rev. B 24 (1981) 115–130

18. M.H. Kalos, D.Levesque, and L.Verlet: Helium at zero temperature with hard-sphere and other forces. Phys. Rev.A 9 (1974) 2178–2195

19. M.H. Kalos and P.A. Whitlock: Monte Carlo Methods Volume I: Basics. JohnWiley, New York (1986).

20. E. Kim and M.H.W. Chan: Observations of Superflow in Solid Helium. Science305 (2004) 1941–1944

21. E.Kim and M.H.W. Chan: Probable observation of a supersolid helium phase.Nature 427 (2004) 225–227

22. W.L. McMillan: Ground state of liquid 4He. Phys. Rev. 138 (1965) A442–A45123. N.Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E.Teller: Equa-

tion of state calculations by fast computing machines. J.Chem. Phys. 21 (1953)1087–1092

24. F.Pederiva, G.V. Chester, S.Fantoni, and L.Reatto: Variational study of vacanciesin solid 4He with shadow wave functions. Phys. Rev. B 56 (1997) 5909-5917

25. F.Pederiva, A.Ferrante, S.Fantoni, and L.Reatto: Homogeneous nucleation of crys-talline order in superdense liquid 4He. Phys. Rev. B 52 (1995) 7564–7571

26. E.L. Pollock and D.M. Ceperley: Simulation of quantum many-body systems bypath-integral methods. Phys. Rev. B 30 (1984) 2555–1568.

27. A.Sarsa, K.E. Schmidt, and W.R. Magro: A path integral ground state method.J.Chem. Phys. 113 (2000) 1366–1371

28. K.E. Schmidt, M.H. Kalos, M.A. Lee, and G.V. Chester: Variational Monte Carlocalculations of liquid 4He with triplet correlations. Phys. Rev. Lett. 45 (1980)573–576

29. S.A. Vitiello: Relative stability of hcp and fcc crystalline structures of 4He, Phys.Rev. B 65 (2002) 214516–214520

30. S.A. Vitiello, K. Runge, and M.H. Kalos: Variational calculations for solid andliquid 4He with a “shadow” wavefunction. Phys. Rev. Lett. 60 (1988) 1970–1972

31. S.A. Vitiello and K.E. Schmidt: Optimization of 4He wave functions for the liquidand solid phases. Phys. Rev. B 46 (1992) 5442–5447

32. S.A. Vitiello and K.E. Schmidt: Variational Methods for 4He using a modern he-hepotential. Phys. Rev. B 60 (1999) 12342–12348

33. S.A. Vitiello and P.A. Whitlock: Green’s function Monte Carlo algorithm for thesolution of the Schrodinger equation with the shadow wave function. Phys. Rev.B 44 (1991) 7373–7377

34. P.A. Whitlock, D.M. Ceperley, G.V. Chester, and M.H. Kalos: Properties of liquidand solid 4He. Phys. Rev. B 19 (1979) 5598–5633

35. P.A. Whitlock and R.M. Panoff: One-body density matrix and the momentumdensity in 4He and 3He. Proc. of the 1984 Workshop on High-Energy Excitationsin Condensed Matter, ed. R.N. Silver. LA-10227-C Vol II (1984)

36. P.A. Whitlock and R.M. Panoff: Accurate momentum distributions from compu-tations on 3He and 4He. Can. J. Phys. 65 (1987) 1409–1415

quantum monte carlo simulations of solid 4whitlock/lsscs06pub.pdf · quantum monte carlo...

Documents